Partial Derivatives
Derivatives with respect to one variable of several.
Key idea
Derivatives with respect to one variable of several.
In depth
For f(x, y), the partial derivative ∂f/∂x differentiates with respect to x while treating y as constant. It is the slope of the surface in the x direction.
Worked example
f = x²y + 3y: f_x = 2xy, f_y = x² + 3.
Formula
Real-world uses
- Measuring how a house price responds to size while location is fixed in economics
Common mistake
Differentiating the other variable as well.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Partial Derivatives?
Derivatives with respect to one variable of several. It is a topic at the expert level in Calculus.
What is the formula for Partial Derivatives?
The key formula is ∂f/∂x.
Where is Partial Derivatives used in real life?
Real-world examples: Measuring how a house price responds to size while location is fixed in economics.
What should I learn before Partial Derivatives?
Make sure you are comfortable with Vectors and Space Curves, the previous topic in Calculus.
What is a common mistake with Partial Derivatives?
Differentiating the other variable as well.
How can I practise Partial Derivatives?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Partial Derivatives?
Move on to Gradient next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.